A+Mathematics+Learning+Profile+of+an+Individual+Child


 * Field Report 3: A Mathematics Learning Profile of an Individual Child**

A student’s mathematical reasoning and understanding is crucial to their development of mathematical concepts. Students need to be supplied inquiry based tasks frequently thorough the year to exercise their knowledge of problem solving strategies while experimenting with new ones. The benefit of inquiry based tasks is students receive opportunities to apply what they have learned in instruction and formulate new methods to solving problems. Furthermore, students develop reasoning skills and learn that justifying their answers is just as, if not more important, than solving the problem themselves. For educators, supplying students with inquiry based tasks, allows them to learn more about the way in which the students think, with what strategies they are secure and insecure in implementing, and overall, if the child understands the big idea conveyed through the problem. These tasks, the students’ responses, and the teacher’s observations, all assist in compiling the child’s mathematical profile similar to the way in which I constructed one of my students from my cooperating classroom.

Mrs. Cox’s class had just finished up their multiplication and division units of study, and therefore, I felt the time was best to implement the inquiry based group lesson plan that was created by myself, Erin McGrady, and Michelle Lapoint. I chose this task because it required students to have a firm foundation of multiplicative concepts as well as for students to know their multiplication facts. The rational for this lesson was to provide students with the opportunity to explore their mastered concepts of multiplication, while producing alternative and unique strategies for solving the problem. My objectives for the students were parallel to the rational as I expected students to activate their knowledge of numerical operations in multiplication as well as experiment with different problem solving strategies to solve for the total number of lawns Ryan will mow in the different situations described in the different steps of the word problem. I was also looking for students to provide reason to how they approached the problem and the methods they used for solving. The lesson was implemented on March 31, 2011 from 1:45 to 2:35 to the whole class.

I began the lesson by introducing the students to a strategy of multiplying with money. The strategy, as referred to me by my cooperating teacher, is to remove the zeros, decimal point and dollar sign from the dollar amount and then multiply regularly. Afterwards, add back on to the equation the dollar sign in the front and the decimal point and zeros on the back. We practiced three problems together and then read the prompt, underlining what was possible important information, and then the problems that they had to solve together. From there, they were given the charge to work independently at their desks using whatever methods and tools (including tiles, extra paper, etc.) that they found most helpful in solving the problem. Students were also instructed to write in full sentences how they solved the problems and to show all of their mathematical computations. They were given approximately 20 minutes to work on the problems and if they finished before time was up, they were encouraged to try some of the extension problems of the backside of the prompt. I walked around answering questions and interviewing a particular student, Lindsey, about her method of solving which will be further developed later in the report. After time was up, we reassembled on the carpet to share the different methods used to solve each on the problems, one at a time. I concluded the lesson by asking students if they enjoyed this problem and what concepts they learned and/or exercised through solving and discussing.

Lindsey is a proficient math student whose understanding of multiplication I found to be interesting enough to study further. Lindsey has a firm understanding of foundational concepts of multiplication including making arrays and translating them into multiplication sentences. This was observed in her rapid but accurate solving of another inquiry based problem I implemented two weeks ago of finding the number of pant/ sock combinations you could make with four different colors of each pants and socks. Lindsey and a partner were able to organize the combinations systematically into a four by four array, which she informed me of, and then further explained that because the array was four by four, that there were 16 combinations since four times four was 16. She also displayed her understanding of arrays on her multiplication examination and in her reflection she noted it as her strength within the unit. She also understands the commutative property in multiplication, multiplication terminology including the product and factors, and writing multiplication sentences from a picture or story. Interestingly, as proficient as Lindsey is in the foundation of multiplication, she struggles with more complex operations and in memorizing facts.

Lindsey’s weaknesses within the multiplication unit involve skills that require high orders of thinking and some rote memorization in terms of reciting times tables. It was interesting to see that although she is knowledgeable on writing a multiplication sentence from a picture or story, she cannot reverse this process. This premature skill is evident on her multiplication assessment when she writes the story of “Nine flowers were in a garden. 3 had yellow petals. How many flowers had pink petals?” for 3x9. Her thinking is focused on subtraction rather than the difference as her story instructs readers to find the difference in the color of petals. She also has difficulty understanding multiplication in terms of addition. On her exam 8x6 was represented as 6+6+6+6+6+6+6+6=48 instead of the number eight added together six times. Another one of Lindsey’s struggles is factors. She was placed in Mrs. Cox’s lab of identifying the factors of numbers two through twelve but relied on the pattern she was coloring in to identify the factors. With this, her most significant problem is the memorization and comprehension of times table facts. She notes this weakness in her reflection completed at the end of the unit and also makes it her goal to memorize fast facts. Her struggle is evident in her exam when she notes “9x0 > 1x4” and on any problem that involves recall of facts because she solves them through repeated addition.

Within instruction, Lindsey understands the basic ideas when presented to the whole group. Here, directions are explicit and she, as well as the rest of the class, are participating in guided practice by solving the problems together. Transitioning to independent work, she needs supplemental assistance as she has difficulty applying strategies that she learns in whole group instruction independently. She almost second guesses herself in a way, and thus needs scaffolding to guide her through the correct processes. She is cognizant and attentive during whole group instruction thus it is difficult to understand why she fails to transcend the strategy taught for in whole group to a similar one in independent practice. She needs assistance to clarify thinking and create individual strategies suited to her understanding of mathematics.

Lindsey’s performance on the summer jobs task was surprising considering her past performance on applying her knowledge of facts. The prompt and questions went as such: Ryan is going to earn a little money this summer by mowing lawns in his neighborhood. He plans to charge $5.00 per yard and has his schedule worked out for the first two weeks of summer. In the first week he will mow 10 lawns and he has planned to mow 7 lawns in the second week. 1. A. How much will Ryan earn in the first week? B. How much will Ryan earn in the second week? C. How much money will Ryan earn in the two weeks? 2. A. If Ryan continues this pattern every two weeks, how much will he make for one month? (with 4 weeks in a month) B. How much will Ryan make in a whole summer if the summer is 3 months? For the first question, 1A. , Lindsey applied her knowledge of ten facts, and removed the one from the ten and put a five in its place to get fifty. The multiplication sentence as she internalized was five times ten (5x10) and five times ten equals fifty (5x10=50) hence her first answer as fifty (50). To solve the second question, 1B, Lindsey used the most interesting method, that I had her share it with the class when we were discussing strategies. Her method for solving what she notes as “7x5” was to subtract ten from seven to get three (10-7=3). With the three, she multiplied it by five to get fifteen (5x3=15). With the fifteen, she then subtracted it from the fifty and got thirty-five (50-15=35) which is also the product of seven times five. To answer the final question in part 1, Lindsey added the amount from week one, which was fifty dollars, and the amount from week two, thirty five dollars to get a total of eighty-five dollars. For my second set of questions, Lindsey solved 2A. By adding eighty-five and eighty five together (85+85) because she knew the pattern was just the week one and two combined and repeated get the total amounts between weeks, which was four as I designated. She also noted that you could solve 85+85 by doing eighty-five times two (85x2) because “if $85 is for 2 weeks and I need 4 weeks which is 2x 85 or 85 +85=170 dollars”. She used the amount she solved for in one month to answer problem 2B. She writes “170+170+170” or “170x3” because for one month, Ryan only made 170 dollars, but she is looking to solve for the amount he made in 3 months. She comes to the correct answer though of 510 dollars.

After analyzing Lindsey’s methods of solving the Summer Jobs problem, I was pleased to see that she had an understanding of the mathematical big idea of multiplication, as well as using some innovative strategies in solving. Multiplication in terms of times table facts, writing mathematical sentences from a story problem, and general problem solving were the big mathematical ideas I was looking for all students and in particularly Lindsey to realize and utilize in solving. She demonstrates her knowledge of the big idea in her uses of multiplication as a strategy in answering the problems. This is evident in the work that she shows and her explanation for solving. More so, her using multiplication in unconventional ways when she solves for the amount of money Ryan made in the second week of summer if he mowed seven lawns at five dollars each, supports her understanding of the big idea as she is capable of manipulating it. Instead of simply multiplying seven times five (7x5), she subtracts ten from seven to get a difference of three (10-7=3) and them multiplies that difference from five (3x5). With the product of fifteen, she subtracts it from fifty (50-15), paralleling her subtraction of the ten minus seven (10-7) and comes to the conclusion of thirty five (50-15=35) which is also seven times five (7x5). Also noted from her performance and understanding of the big idea, was that although she knew to apply multiplication as the method for solving, she was ill-equipped with carrying through with the procedure. When it came to multiplying the larger number by smaller ones, she used repeated addition to find the answer instead of using multiplication. She set up the problem correctly to solve multiplicatively, but then failed to demonstrating her lack of times table knowledge. Although, with her use of repeated addition, it is deduced that she possesses keen and accurate addition skills with large numbers. Finally, she comprehends an even more specific big idea of multiplying with money as she applies the strategy instructed upon before exploring the task at hand. She removed the zeros, decimal point, and dollar sign, multiplied regularly and then adds them all back into the appropriate locations in the product. Through Lindsey’s work, application, and reasoning for using of multiplication on almost all the problems, she therefore proves her understanding of the big mathematical idea of multiplication.

Inquiry based tasks are essential elements to mathematical instruction as students develop reasoning skills, problem solving strategies, and an understanding of mathematical big ideas through them. Just as the summer jobs problem provoked students to think critically over methods of solving and explanations for doing so, it also fostered the mathematical idea of multiplication and its applications to problem solving. Lindsey was a prime example of demonstrating the understanding of the big idea as she used a variety of multiplicative techniques in solving the problems. She was able to manipulate multiplication, write mathematical sentences from it, and her knowledge of facts and repeated addition as multiplication assisted in her solving. Overall, inquiry based tasks, an understanding and development of the child’s mathematical profile and conveying of a big mathematical idea are all components of meaningful mathematics instruction.

Child's Work

Summer Jobs Lesson Plan